Theorem cbviunv

Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003) Add disjoint variable condition to avoid ax-13 . See cbviunvg for a less restrictive version requiring more axioms. (Revised by GG, 14-Aug-2025)

	Ref	Expression
Hypothesis	cbviunv.1	|- ( x = y -> B = C )
Assertion	cbviunv	|- U_ x e. A B = U_ y e. A C

Proof

Step	Hyp	Ref	Expression
1		cbviunv.1	|- ( x = y -> B = C )
2	1	eleq2d	|- ( x = y -> ( z e. B <-> z e. C ) )
3	2	cbvrexvw	|- ( E. x e. A z e. B <-> E. y e. A z e. C )
4	3	abbii	|- { z | E. x e. A z e. B } = { z | E. y e. A z e. C }
5		df-iun	|- U_ x e. A B = { z | E. x e. A z e. B }
6		df-iun	|- U_ y e. A C = { z | E. y e. A z e. C }
7	4 5 6	3eqtr4i	|- U_ x e. A B = U_ y e. A C